A Look at Musical Instrument Sounds

Background on Fourier Analysis/Synthesis

· Musical instruments have different tonal qualities, even when playing the same note. This is due to different “harmonics” present in the instruments signal.

· Playing the same note on any instrument will produce the same fundamental frequency (the lowest frequency sinusoidal component). However, sinusoids of integer multiple frequencies of this fundamental are also present (the harmonics). You hear the sum of these harmonics and the fundamental. Your brain recognizes the fundamental frequency and associates that with the “note” you hear. The harmonics give the note its “character.”

· Each instrument has a certain set of harmonic strengths unique to that instrument. A flute has relatively weak or soft harmonics and produces nearly a perfect sinusoid at the fundamental frequency. A muted trumpet, on the other hand, is rich with strong harmonics giving it a “growl” sound.

· What is interesting is that any waveform (signal shape) can be expressed as the sum of sinusoidal signals of various strengths and frequencies. This is what we refer to as the frequency content of a signal. This concept is credited to Fourier and determining the frequency content of a signal is called Fourier analysis. Building a signal by adding harmonics is called Fourier synthesis.

· A “synthesizer” is the name for a keyboard that constructs its sounds by adding sinusoids (performs Fourier synthesis). Most keyboards today are sampling keyboards and get their sounds by digitally recording the actual instruments.

· Watch how a saw-tooth wave shape can be created (synthesized) by adding sinusoidal harmonics to a fundamental sinusoidal signal

clear; close all; clc

figure(1)

t=linspace(0,10,500);

out=zeros(size(t));

for k=1:10

sign=(-1)^(k-1);

out=out+sign*(1/k)*sin(k*t); % add on the k-1’th harmonic

plot(t,out)

xlabel('time (sec) ');

ylabel('x(t) ');

title('Saw-tooth Wave Formed with Sinusoidal Harmonics');

grid

axis([0,10,-2,2]);

if k==1

S=sprintf('Fundamental Frequency');

else

S=sprintf('%.0f Harmonics',k-1);

end

text(4.25,1.25,S);

M(k)=getframe;

disp('Hit Any Key to Add Next Harmonic');

pause

end

movie(M,3,10);

· Here is a demonstration of how a square-wave can be synthesized by adding sinusoidal harmonics in just the right proportion. This page was created by Paul Hartke when he was an undergraduate Electrical Engineering Student at UD.

Musical Instrument Signals

· Now that you know a thing or two about harmonics, lets look at the waveforms produced by real instruments. These sounds have been recorded from my KORG 01/W Pro Sampling Keyboard.

clear; close all; clc

[flute,fs]=wavread('flute.wav');

[tenor,fs]=wavread('tenorsax.wav');

[trump,fs]=wavread('mutedtrumpet.wav');

· Let’s look at, and listen to, the flute signal

figure(1)

N=length(flute);

t=linspace(0,N/fs,N);

plot(t(7000:7500),flute(7000:7500))

xlabel('time (sec) ');

ylabel('relative strength');

title('Small Portion of Flute Signal');

soundsc(flute,fs);

· When we look at an audio signal in time and in frequency, this is called a spectrogram. The spectrogram you are about to see is color-coded. Red means strong energy content, and blue is little energy. You should be able to see the fundamental frequency energy (lowest red line) going up and down the scale over time and the harmonics go right along with it (their frequencies are always integer multiples of the fundamental).

% flute

figure(1)

specgram(flute,1000,fs,1000)

soundsc(flute,fs);

· Try this with the tenor sax and muted trumpet

% tenor sax

figure(1)

specgram(tenor,1000,fs,1000)

soundsc(tenor,fs);

% trumpet

figure(1)

specgram(trump,1000,fs,1000)

soundsc(trump,fs);

· You can see how many strong harmonics there are with the muted trumpet. Let’s look at a piece of the signal (amplitude vs. time) and listen again. The rough shape of the waveform indicates the presence of many strong harmonics (the less like a pure sinusoid, the more harmonics it must have).

figure(1)

N=length(trump)

t=linspace(0,N/fs,N);

plot(t(1000:2000),trump(1000:2000))

xlabel('time (sec) ');

ylabel('relative strength');

title('Small Portion of Trumpet Signal');

soundsc(trump,fs);